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Full text of "Analytical Mechanics"

MOTION OF A PARTICLE 107 rections of the tangent and the normal and then dropping the vector notation we obtain FT = mo, (VII) and • Fn = -m~. (VIII) P The negative sign in equation (VIII) states that the normal component of the resultant force, and consequently the resultant force itself, is directed toward the concave part of the path. The last two equations may be obtained directly from (VI) by considering them as the force equations for special cases of motion. Thus when the path of the moving particle is a straight line p = oo, and consequently F = mi). (VII;) On the other hand when the particle moves with a constant speed v = 0, and therefore F= ~m~' (VIHO If in addition the radius of curvature of the path does not change, that is, if the particle moves in a circle with a constant speed, then 2,2 F=-m7> (vm'o where r is the radius of the circle. The following is a useful set of component-force equations obtained by splitting equation (V) into three component equations which correspond to the directions of the axes of a rectangular system: F, = nti, Fy =. my, (IX) Fz = mz. Equations (IX) emphasize the fact that the component of the resultant force along any direction equals the product of